By [1]}, for \(\mathcal {L}^1\) -a.e. \(x_n\in D_{x^{\prime }}^{h}\) , \(\nabla f(x^{\prime },x_n)\)  is equal to a constant. By [1]}, for \(\mathcal {L}^1\) -a.e. \(x_n\in \tilde{D}_{x^{\prime }}^{h}\) , \(\nabla Sf(x^{\prime },x_n)\)  equals the same constant as \(\nabla f(x^{\prime },x_n)\)  for \(x_n\in D_{x^{\prime }}^{h}\) . Thus by \(|D^h_{x^{\prime }}|=|\tilde{D}^h_{x^{\prime }}|\) , (REF ) is established. By (REF ) and the arbitrariness of \(h\in D_1\) , (REF ) is established. By (REF ) and Fubini's theorem, (REF ) is established.
